{"id":"2d460d85-ae18-4032-a4ef-57bf25bcd9d6","arxiv_id":"2505.19066","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Oscillatory couplings with frequency above the heavy field mass remove the Boltzmann suppression of cosmological collider signals and produce new scale-dependent bispectrum shapes, with axion monodromy as a concrete realization.","lead":"This paper computes how heavy particles during inflation can leave large fingerprints in the primordial bispectrum when their couplings oscillate faster than the particle mass. It shows these resonant collider signals can overcome the usual exponential suppression and may reveal heavy string-theory moduli in cosmological data.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The x1-constant trick in Sec. 3.1 is the load-bearing technical step; without a proof or numerical check that restoring x1=kη1 reproduces the exact bulk integral, the claimed resonance enhancement of f_NL is not yet established.","rationale":"The reader's weakest assumption pinpoints the same step. I considered alternatives—the sketched weight-shifting operator, the folded-limit resummation borrowed from Ref. [23], and the string-model parameter choices—but these are either plausibly correct or model-dependent. The x1 trick is the unique technical keystone for both the scalar seed and the enhancement factors: every new result, from Eq. (3.7) to the squeezed-limit bispectra of Sec. 4.2, inherits it. The paper provides partial validation by showing the α=0 limit reduces to known results and the contact example in Sec. 2.3 is exact, but the exchange computation lacks an independent check against the defining bulk integral. The proposed numerical test is decisive because it compares the full analytic result to the exact bulk integral that defines the correlator. I therefore keep the verdict CONDITIONAL/UNCHANGED: accept only after the test is passed, or after a proof of the trick's validity is supplied.","tokens_in":36420,"tokens_out":7071,"duration_ms":66939,"concrete_test":"Evaluate the mixed-propagator integral (3.1) numerically for a benchmark, e.g., μ=10, α=15, η1 such that kη1=10^-2, using the exact Hankel mode functions (2.4), without any x1 substitution; compare the soft-limit K̂(kη,x1) to Eqs. (3.6)-(3.7) after restoring x1=kη1. Then compute the scalar seed I_a from Eq. (3.20) by direct numerical integration for u=0.1, 0.5, 0.9 and compare with the analytic solution (3.23)+(3.25) with x1,x2 restored. Agreement at the <0.1% level would validate the trick; O(1) disagreement would invalidate the derived enhancement factors and the central claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Sec. 3.1 the authors handle a time-dependent coupling by defining x1≡kη1, treating it as a k-independent constant when deriving the boundary differential equation (3.5) and when solving for the mixed propagator, and only restoring x1=kη1 at the end. This is legitimate only if the solution's x1-dependence is such that the additional terms from differentiating x1 with respect to k vanish or are subleading; no such proof is given, and the paper does not check the restored solution against the original bulk integral. The enhancement factors EB1 and EB2 in Eqs. (4.6)-(4.7), and hence the predicted fNL ~ 10-100 in Sec. 5.4, are extracted from the coefficients Ca± in the homogeneous solutions (3.25)-(3.26), which inherit the x1-trick. If the trick misses k-dependent terms, the scale dependence and the resonance enhancement could be quantitatively wrong, even though the power-law form of the oscillatory coupling makes a cancellation plausible. A direct numerical check is therefore required before the central claim is accepted.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops a boundary (bootstrap) computation of massive-exchange three-point correlators with periodically oscillating couplings, extending the boostless bootstrap to backgrounds with a discrete shift symmetry. The authors derive boundary differential equations for the mixed propagator of the quadratic φ̇σ vertex (Sec. 3.1) and for the primary scalar seeds with one (Sec. 3.2.1) or two (Sec. 3.2.2) oscillatory couplings, solve them analytically in hypergeometric series, and obtain inflaton bispectra by weight-shifting (Sec. 3.3). The central claim is that resonances between the Bunch-Davies vacuum, the massive-mode oscillations, and the oscillatory coupling (frequency ω = αH) remove the Boltzmann suppression e^{-πµ} for m ≫ H when ω ≳ m; the enhancement factors E_B1, E_B2 (Eqs. (4.6)–(4.7)) and the new two-frequency bispectrum shapes are computed in full. These results are then applied to a two-field axion-monodromy model with a heavy modulus (Sec. 5), where the authors argue that the modulus cannot be integrated out and that the squeezed bispectrum can reach f_NL ~ O(10–100).","tokens_in":36660,"tokens_out":44008,"duration_ms":350858,"significance":"If correct, this is a substantial advance: it provides the first full analytical bootstrap solution for cosmological-collider signals with explicitly scale-dependent (resonant) features, renders the approximate analysis of Ref. [55] exact at the level of full kinematic shapes, and identifies a concrete UV-motivated scenario in which the conventional single-field EFT fails because a heavy modulus is continuously excited. The manuscript contains several genuine internal checks that give moderate confidence in the technical core: the α = 0 limits reduce to the known results of Ref. [17]; the contact example of Sec. 2.3 is verified against the explicit bulk integral; the enhancement-factor asymptotics (4.8) reproduce the approximate results of Ref. [55]; and the detailed ODE derivation in Appendix B is explicit. The principal weaknesses are the unjustified 'x1 trick' (Sec. 3.1), the stated-but-unproven weight-shifting operator for the φφ̇σ vertex (Sec. 3.3), and the borderline parameter benchmarks used for the headline f_NL estimates (Sec. 5.4).","major_comments":[{"comment":"The 'x1 trick' of Sec. 3.1 is the load-bearing technical step of the paper: x1 ≡ kη1 is treated as a k-independent constant in deriving and solving the boundary differential equations, and k-dependence is restored only at the end. The manuscript does not justify this step. It is not a priori innocuous: after restoring x1 = k3η1, the scalar seeds no longer satisfy the displayed ODEs with respect to k3, since ∂_{k3} also acts on x1 and generates O(α/k3) terms that are absent from the derivation. For the couplings actually treated (cos(α log...), a sum of two power laws) the step can be shown to be exact: for a power-law coupling the x1-dependence of every component is a multiplicative phase (kη1)^{∓iα}, which commutes with O_η, and the soft-limit boundary conditions (3.6)–(3.7) fix both homogeneous coefficients, so the ODE solution with x1 restored coincides with the bulk integral. This argument is not given, and the claim that the method 'can be generalized to deal with arbitrary time-dependent couplings' is not supported. I request a proof of exactness for power-law couplings (covering the cos case by linearity), or alternatively a numerical comparison of (3.9)/(3.20) with direct evaluation of the corresponding bulk integrals, before the enhancement factors (4.6)–(4.7) and the f_NL estimates are treated as established.","section":"Sec. 3.1, Eqs. (3.3)–(3.10), (3.21)"},{"comment":"The weight-shifting operator W^{φφ̇σ} for the cubic vertex φφ̇σ is stated in Eqs. (3.29)–(3.31) ('we simply notice that its bulk integral can be written into the following form') rather than derived, in contrast to the (∂µφ)²σ operator (3.27), which is quoted from the literature. This operator underlies the bispectrum template (3.30) and the dominant axion-monodromy signal (5.37)–(5.38). Please provide the derivation, including the action of ∂_η on the external K^φ factors and the commutation with the oscillatory phase α2 log(k3η/x2), or verify (3.29)–(3.31) against a direct evaluation of the bulk integral for representative kinematics.","section":"Sec. 3.3, Eqs. (3.29)–(3.31)"},{"comment":"The headline estimates f_NL ~ O(10)–O(100) rest on benchmarks that sit at or beyond the edge of the model's own consistency conditions. The choice ε0 M_Pl²/(2Λ²) = 1 used for Fig. 10 and the 'O(10)' statement gives Φ̇/Λ = 2H, which violates the last condition of Eq. (5.6), 'Φ̇/Λ ≪ H'. The third condition of (5.6), 'Λ, Λ̃ ≫ Φ̇√αH', is also typeset ambiguously (multiple readings are possible) and should be stated precisely. As written, the reader cannot tell whether the f_NL ~ 10–100 numbers are attainable inside the region where the two-field description is self-consistent. In addition, the two displayed lines of (5.38) are inconsistent: with |F^{φφ̇σ}(µ,α)| = |5+i(µ+α)| ≈ α for α ≫ µ, the first line gives a coefficient −1/4 (not −1/2); the replacement |F| ≈ 2α is valid only for α ≈ µ. Please fix the asymptotics and the benchmark bookkeeping and report the maximum f_NL within the allowed region.","section":"Sec. 5.4, Eqs. (5.6), (5.36)–(5.40), Fig. 10"}],"minor_comments":[{"comment":"In the Introduction, the sentence 'ω is larger than the massive of the heavy field' should read 'larger than the mass of the heavy field'.","section":"Introduction"},{"comment":"The phrase 'the slow-roll field velocity ˙Φ1/2 ≃ 60H' is ambiguous; please specify whether the square root of the velocity or another combination is intended.","section":"Sec. 5.1"},{"comment":"In the text following Figure 1, the sentence 'the seed function in Figure 1a is featureless' refers to Figure 1a twice; the second reference should presumably be to Figure 1b.","section":"Sec. 3.2.2 / Fig. 1"},{"comment":"The amplitude relations δn_α = −λ0λα|E_P1| and δn_{2α} = −λα²|E_P2| appear to miss a factor of 2 relative to the explicit c.c. in (4.4); please state the convention for δn_α, since Eq. (5.36) inherits it.","section":"Sec. 4.1, Eqs. (4.4)–(4.5)"},{"comment":"The evaluation leading to the soft-limit coefficients A±(x1) in Eq. (3.7) is stated without derivation; please include the computation of the k → 0 limit of the integral (3.1) or provide an explicit reference.","section":"Sec. 3.1, Eq. (3.7)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits the scope of this journal well: the technical core is EFT/amplitude-style bootstrap theory and the motivation is string cosmology. The companion letter [61] is clearly complementary, and no novelty issue arises. My main message to the editor: the two biggest technical steps (the x1 trick and the weight-shifting operator for φφ̇σ) are presented as facts rather than derivations. I checked the x1 trick by hand for the power-law components of the cos coupling and found it to be exact because the x1-dependence is a multiplicative phase; hence this is a presentation gap that a short proof can close, not a demonstrated error. The factor-of-2 inconsistencies in (4.5)/(4.4) and (5.38), and the benchmark-validity issue with respect to (5.6), are also fixable. I therefore view major_revision as the appropriate verdict rather than reject. I would additionally ask the authors to state, for the model of Sec. 5, the maximal f_NL consistent with all conditions in (5.6), since the abstract's 'detectably large' signal claim is parameter-dependent."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper: it is the first full bootstrap treatment of massive-exchange bispectra with oscillatory couplings, and it gives explicit analytic templates showing the resonance mechanism can lift the Boltzmann suppression for m≫H. The string-inspired axion-monodromy application is a genuine bonus: it connects the formalism to a concrete UV model and produces fNL of order 10–100 without obviously violating power-spectrum constraints.\n\nWhat is actually new: the boundary differential equations (3.21) for the scalar seed with two oscillatory vertices, the closed-form homogeneous/particular solutions, and the derivation of the bispectrum shapes and enhancement factors EB1, EB2. The Appendix B derivation is careful. The α=0 limit correctly reproduces known cosmological collider results, and the enhancement factors match the approximate stationary-phase results of Ref. [55]. That is real evidence the computation is not just a repackaging.\n\nThe soft spots are manageable but real. The largest is the x1≡kη1 trick in Sec. 3.1: treating x1 as a k-independent constant when deriving and solving the differential equations, then restoring k-dependence at the end. The paper offers no proof that the extra k-derivatives of x1 are negligible, and no numerical comparison of the restored solution against the exact bulk integral. I believe the result is probably correct—the power-law form of the coupling makes the cancellation plausible—but the scale dependence of the templates and the quoted enhancement factors rest on this step. A direct numerical check would settle it, and the authors should add one.\n\nTwo smaller issues: the weight-shifting operator for the ϕ̇ϕσ vertex (3.31) is stated rather than derived, and the folded-limit power-spectrum resummation is imported from Ref. [23] without an independent derivation. Also, the claim that ω≳m is 'relatively common' in string compactifications is asserted rather than demonstrated; the model section would be stronger with a survey or a concrete compactification example.\n\nOverall, the central physics mechanism (resonance with ω≳m overcomes Boltzmann suppression) is credible and well supported by the analytic structure, including the α=0 limit and agreement with earlier approximate results. The x1 trick is a gap, not a fatal flaw. This paper deserves a serious referee; I would ask the referee to focus on Sec. 3.1 and on the model parameter scan. I would cite it if I work on cosmological colliders or resonant non-Gaussianity.\n\nRecommendation: send to peer review.","headline":"A solid bootstrap computation for resonant cosmological colliders whose central caveat is the unproven x1 trick; worth refereeing.","tokens_in":37211,"tokens_out":2737,"would_cite":true,"duration_ms":25994,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["98.80.Cq","98.80.Es","11.25.-w"],"model":"deepseek-v4-flash","headline":"Periodically varying couplings during inflation can amplify the cosmological collider signal of heavy particles, replacing the Boltzmann suppression with a resonance enhancement when the oscillation frequency is at least the particle mass.","keywords":["cosmological collider","resonant non-Gaussianity","cosmological bootstrap","primordial bispectrum","axion monodromy","heavy fields during inflation","oscillatory couplings","single-field EFT"],"falsifier":"Numerically evaluating the bulk in-in integral for the mixed propagator and the scalar seed $\\hat I(u,x_1)$ at $\\alpha\\ge\\mu$ and comparing the resulting $k$-running with the boundary-bootstrap solution would settle the $x_1$ trick; disagreement at the level of the enhancement factors $E_{B1},E_{B2}$ would refute the paper's central resonance claim.","tokens_in":36169,"feed_emoji":"🌌","tokens_out":7842,"duration_ms":63837,"temperature":0.7,"pith_summary":"The paper argues that in inflationary models with a discrete shift symmetry, periodically oscillating couplings act as a resonant pump for heavy fields, so masses far above the Hubble scale no longer make particle signals exponentially invisible. It derives boundary differential equations for the massive-exchange three-point functions and solves them analytically for any kinematics. It finds that when the oscillation frequency satisfies $\\omega \\gtrsim m \\gg H$, the Boltzmann factor $\\mathrm{e}^{-\\pi m/H}$ is replaced by a much milder suppression, and new non-Gaussian shapes appear that combine resonant features with the collider signal. Applied to axion monodromy inflation with a heavy modulus, the result implies that the modulus cannot be integrated out and can produce $f_{\\mathrm{NL}}$ of order 10 to 100 in the squeezed limit.","feed_headline":"Oscillating couplings rescue heavy-particle signals in inflation","feed_subtitle":"A boundary bootstrap shows heavy fields far above the Hubble scale can still leave detectable non-Gaussianity.","key_machinery":"The engine is the boundary bootstrap for massive exchanges with oscillatory couplings, built from the mixed propagator $\\hat K_\\pm(k\\eta,x_1)$ and the scalar seed $\\hat I(u,x_1)$. The mixed propagator converts a massive field into the inflaton through a time-dependent quadratic vertex and satisfies an inhomogeneous equation of motion; by defining $x_1\\equiv k\\eta_1$ and treating it as a constant while solving, the equation can be traded for $k$-derivatives and solved systematically. The scalar seed obeys a second-order boundary differential equation in $u=k_3/k_{12}$ whose operator is modified by the oscillatory frequency $\\alpha$, with sources fixed by the oscillatory couplings. Homogeneous and particular solutions encode the collider signal and the resonant features, and weight-shifting operators then map the seed to the inflationary bispectrum.","core_discovery":"The paper's central claim is that resonance among the Bunch-Davies vacuum oscillations $\\mathrm{e}^{\\mathrm{i}k\\eta}$, the massive-field oscillations $\\mathrm{e}^{\\pm\\mathrm{i}mt}$, and a coupling $\\cos(\\omega t)$ replaces the Boltzmann suppression for heavy fields: the collider signal is no longer exponentially small once $\\omega \\gtrsim m \\gg H$. In the bootstrap solution this shows up in enhancement factors $E_{B1}$ and $E_{B2}$ that grow from $\\mathrm{e}^{-\\pi\\mu}$ to roughly $1/\\sqrt{\\mu\\alpha}$ when the coupling frequencies $\\alpha_1,\\alpha_2$ reach the mass parameter $\\mu$, with a softened suppression $\\mathrm{e}^{-\\pi(\\mu-\\alpha_1)}$ in between. The same mechanism means a heavy modulus in axion monodromy inflation is continuously excited by the oscillating axion background; the single-field EFT breaks down, and the squeezed bispectrum can carry $f_{\\mathrm{NL}}$ of order 10 to 100. The paper derives the full analytical shape of these correlators rather than only the squeezed limit.","pith_inferences":["The $x_1$ trick is the load-bearing approximation; a natural next test is to repeat the boundary solution for a coupling with a finite-width or localized feature to see whether the resonant enhancement survives beyond the exact power-law form.","The same resonance mechanism should operate in other oscillating inflationary systems, such as periodic turns in multi-field inflation or oscillating sound speed, where analogous boundary equations could be written down and checked for enhancement.","If the combined template is searched in data, the ratio of the collider frequency to the resonant running frequency would directly measure $\\mu/\\alpha$, i.e. the ratio of the heavy mass to the oscillation frequency, and a nondetection would set model-independent bounds on $\\Lambda$ and $b_*$ within this class of models."],"forward_implications":["Heavy fields with $m\\gg H$ no longer have to be exponentially invisible: whenever the coupling oscillates at frequency $\\omega\\gtrsim m$, the squeezed bispectrum carries collider oscillations at order-one amplitude rather than $\\mathrm{e}^{-\\pi m/H}$.","The single-field EFT of axion monodromy inflation is not universally valid: in the regime $\\omega\\gtrsim m$ the heavy modulus is continuously excited, so integrating it out misses the dominant bispectrum.","The primordial bispectrum acquires a new family of shapes that superimpose scale-invariant collider oscillations with scale-dependent resonant running, giving templates with a clear frequency structure for CMB and large-scale-structure searches.","For natural model parameters such as $b_*\\sim 0.1$, $\\epsilon_0 M_{\\mathrm{Pl}}^2/2\\Lambda^2\\sim 1$, and $\\alpha\\lesssim 400$, the model predicts $f_{\\mathrm{NL}}$ of order 10 to 100 while keeping the oscillatory power-spectrum correction within the Planck bound $\\delta n\\lesssim 0.05$."],"supporting_citations":[{"why":"Proposed the classical cosmological collider with oscillatory couplings and estimated the softened Boltzmann factor that this paper derives exactly by bootstrap.","marker":"[55]"},{"why":"Supplies the boostless bootstrap technology (mixed propagator, scalar seed, weight-shifting) that the paper adapts to time-dependent couplings.","marker":"[17]"},{"why":"Establishes the cosmological bootstrap for single-field resonant non-Gaussianity, the discrete-shift-symmetry setting the paper extends to massive exchange.","marker":"[25]"},{"why":"Defines the cosmological collider signal and its Boltzmann suppression in the squeezed bispectrum, which the resonance is claimed to overcome.","marker":"[46]"},{"why":"Introduced the $x_1$ trick of keeping $k\\eta_1$ constant while solving boundary differential equations with additional scale dependence.","marker":"[22]"},{"why":"Provides the closed-form scalar-seed expressions used to evaluate the folded limit and the power-spectrum enhancement factors.","marker":"[23]"},{"why":"The short companion letter on the UV sensitivity of axion monodromy whose two-field model and main phenomenology are expanded here.","marker":"[61]"}],"fun_headline_variants":["Resonance overcomes Boltzmann suppression for heavy fields in inflation","Oscillating couplings make heavy moduli detectable in cosmic bispectrum","Axion monodromy inflation boosts heavy-field signals via resonance","How periodic couplings rescue heavy-particle signals from inflation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation relies on treating $x_1 \\equiv k\\eta_1$ as a constant while solving the differential equations and restoring its momentum dependence only at the end, a shortcut whose full validity for general time-dependent couplings is not proven; if it misses scale dependence, the enhancement factors and running change.","fun_headline_variants_meta":{"raw":{"variants":["Resonance overcomes Boltzmann suppression for heavy fields in inflation","Oscillating couplings make heavy moduli detectable in cosmic bispectrum","Axion monodromy inflation boosts heavy-field signals via resonance","How periodic couplings rescue heavy-particle signals from inflation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000238,"raw_usage":{"total_tokens":1536,"prompt_tokens":997,"completion_tokens":539,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":613,"completion_tokens_details":{"reasoning_tokens":467}},"tokens_in":613,"tokens_out":539,"duration_ms":4977,"temperature":1.0,"reasoning_tokens":467,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:20:45.905168+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically evaluating the bulk in-in integral for the mixed propagator and the scalar seed $\\hat I(u,x_1)$ at $\\alpha\\ge\\mu$ and comparing the resulting $k$-running with the boundary-bootstrap solution would settle the $x_1$ trick; disagreement at the level of the enhancement factors $E_{B1},E_{B2}$ would refute the paper's central resonance claim.","supporting_citations":[],"review_version":1}